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The rank of actions on \(\mathbf R\)-trees. (English) Zbl 0835.20038
For \(n\geq 2\), let \(F_n\) denote the free group of rank \(n\). We define a total branching index \(i\) for a minimal small action of \(F_n\) on an \(\mathbb{R}\)-tree. We show \(i\leq 2n-2\), with equality if and only if the action is geometric. We thus recover Jiang’s bound \(2n-2\) for the number of orbits of branch points of free \(F_n\)-actions, and we extend it to very small actions (i.e. actions which are limits of free actions).
The \(\mathbb{Q}\)-rank of a minimal very small action of \(F_n\) is bounded by \(3n-3\), equality being possible only if the action is free simplicial. There exists a free action of \(F_3\) such that the values of the length function do not lie in any finitely generated subgroup of \(\mathbb{R}\). The boundary of Culler-Vogtmann’s outer space \(Y_n\) has topological dimension \(3n-5\).
Reviewer: D.Gaboriau (Lyon)

MSC:
20E08 Groups acting on trees
20F65 Geometric group theory
57M07 Topological methods in group theory
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